Cooperative networks and Hodge-Shapley value

TL;DR

This paper extends the Shapley value concept to cooperative networks using stochastic path integrals and Hodge calculus, broadening its applicability beyond traditional coalition games.

Contribution

It introduces a novel interpretation of the Shapley value as a stochastic path integral and integrates Hodge calculus to generalize cooperative value allocation in networks.

Findings

Extended Shapley value to partial coalition states

Characterized the stochastic path integral with new axioms

Applied Hodge calculus to cooperative network analysis

Abstract

Lloyd Shapley's cooperative value allocation theory stands as a central concept in game theory, extensively utilized across various domains to distribute resources, evaluate individual contributions, and ensure fairness. The Shapley value formula and his four axioms that characterize it form the foundation of the theory. Traditionally, the Shapley value is assigned under the assumption that all players in a cooperative game will ultimately form the grand coalition. In this paper, we reinterpret the Shapley value as an expectation of a certain stochastic path integral, with each path representing a general coalition formation process. As a result, the value allocation is naturally extended to all partial coalition states. In addition, we provide a set of five properties that extend the Shapley axioms and characterize the stochastic path integral. Finally, by integrating Hodge calculus, stochastic processes, and path integration of edge flows on graphs, we expand the cooperative value allocation theory beyond the standard coalition game structure to encompass a broader range of cooperative network configurations.